calculus
posted by katarina .
1. Find the area of the region bounded by f(x)=x^2 +6x+9 and g(x)=5(x+3). Show the integral used, the limits of integration and how to evaluate the integral.
2. Find the area of the region bounded by x=y^2+6, x=0 , y=6, and y=7. Show all work required in #1.
3. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about y=3. Show the integral and give an exact answer.

There are 3 questions here.
You show no work , nor do you tell us where your difficulty is.
I will start you off with #1
we need their intersection points
x^2 + 6x + 9 = 5x + 15
x^2 + x  6 = 0
(x+3)(x2) = 0
x = 3 or x = 2
if x = 3, y = 0
if x = 2 , y = 25
from x = 3 to x = 2, g(x) > f(x), so the effective height
= 5x + 15  x^2  6x  9
= 6  x  x^2
Area = ∫(6  x  x^2) dx from x = 3 to 2
= [6x  (1/2)x^2  (1/3)x^3] from 3 to 2
= (12  (1/2)(4)  (1/3)(8) )  (18  (1/2)(9)  (1/3)(27) )
= 12  2  8/3 + 18  9/2 + 9
= 179/6 
the answer is 125/6 not 179/6 for problem no. 1